\(\int \frac {1}{(a \sin ^2(x))^{3/2}} \, dx\) [5]

Optimal result

Integrand size = 10, antiderivative size = 42 \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\text {arctanh}(\cos (x)) \sin (x)}{2 a \sqrt {a \sin ^2(x)}} \]

[Out]

-1/2*cot(x)/a/(a*sin(x)^2)^(1/2)-1/2*arctanh(cos(x))*sin(x)/a/(a*sin(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3283, 3286, 3855} \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\sin (x) \text {arctanh}(\cos (x))}{2 a \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}} \]

[In]

Int[(a*Sin[x]^2)^(-3/2),x]

[Out]

-1/2*Cot[x]/(a*Sqrt[a*Sin[x]^2]) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[a*Sin[x]^2])

Rule 3283

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[Cot[e + f*x]*((b*Sin[e + f*x]^2)^(p + 1)/(b*f*(2
*p + 1))), x] + Dist[2*((p + 1)/(b*(2*p + 1))), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \sin ^2(x)}} \, dx}{2 a} \\ & = -\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}+\frac {\sin (x) \int \csc (x) \, dx}{2 a \sqrt {a \sin ^2(x)}} \\ & = -\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\text {arctanh}(\cos (x)) \sin (x)}{2 a \sqrt {a \sin ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\left (\csc ^2\left (\frac {x}{2}\right )+4 \log \left (\cos \left (\frac {x}{2}\right )\right )-4 \log \left (\sin \left (\frac {x}{2}\right )\right )-\sec ^2\left (\frac {x}{2}\right )\right ) \sin ^3(x)}{8 \left (a \sin ^2(x)\right )^{3/2}} \]

[In]

Integrate[(a*Sin[x]^2)^(-3/2),x]

[Out]

-1/8*((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x]^3)/(a*Sin[x]^2)^(3/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(34)=68\).

Time = 0.84 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {\sqrt {a \left (\cos ^{2}\left (x \right )\right )}\, \left (\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (x \right )\right )}+2 a}{\sin \left (x \right )}\right ) a \left (\sin ^{2}\left (x \right )\right )+\sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (x \right )\right )}\right )}{2 a^{\frac {5}{2}} \sin \left (x \right ) \cos \left (x \right ) \sqrt {a \left (\sin ^{2}\left (x \right )\right )}}\)	\(70\)
risch	\(-\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{a \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{a \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\)	\(110\)

[In]

int(1/(a*sin(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/a^(5/2)/sin(x)*(a*cos(x)^2)^(1/2)*(ln(2*(a^(1/2)*(a*cos(x)^2)^(1/2)+a)/sin(x))*a*sin(x)^2+a^(1/2)*(a*cos(
x)^2)^(1/2))/cos(x)/(a*sin(x)^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {-a \cos \left (x\right )^{2} + a} {\left ({\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) + 2 \, \cos \left (x\right )\right )}}{4 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sin \left (x\right )} \]

[In]

integrate(1/(a*sin(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*sqrt(-a*cos(x)^2 + a)*((cos(x)^2 - 1)*log(-(cos(x) - 1)/(cos(x) + 1)) + 2*cos(x))/((a^2*cos(x)^2 - a^2)*si
n(x))

Sympy [F]

\[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a*sin(x)**2)**(3/2),x)

[Out]

Integral((a*sin(x)**2)**(-3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (34) = 68\).

Time = 0.46 (sec) , antiderivative size = 314, normalized size of antiderivative = 7.48 \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {{\left ({\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )\right )} \sqrt {-a}}{2 \, {\left (a^{2} \cos \left (4 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right )^{2} + a^{2} \sin \left (4 \, x\right )^{2} - 4 \, a^{2} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a^{2} \sin \left (2 \, x\right )^{2} - 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2} - 2 \, {\left (2 \, a^{2} \cos \left (2 \, x\right ) - a^{2}\right )} \cos \left (4 \, x\right )\right )}} \]

[In]

integrate(1/(a*sin(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)
^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 -
sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) - 1) + 2*(sin(3*x) +
sin(x))*cos(4*x) - 2*(cos(3*x) + cos(x))*sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(
x)*sin(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))*sqrt(-a)/(a^2*cos(4*x)^2 + 4*a^2*cos(2*x)^2 + a^2*sin(4*x)^2 - 4*a
^2*sin(4*x)*sin(2*x) + 4*a^2*sin(2*x)^2 - 4*a^2*cos(2*x) + a^2 - 2*(2*a^2*cos(2*x) - a^2)*cos(4*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (34) = 68\).

Time = 0.36 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.93 \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}}{\sqrt {a} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {2 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \left (x\right )\right )} + \frac {\cos \left (x\right ) - 1}{\sqrt {a} {\left (\cos \left (x\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}}{8 \, a} \]

[In]

integrate(1/(a*sin(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/8*((2*(cos(x) - 1)/(cos(x) + 1) - 1)*(cos(x) + 1)/(sqrt(a)*(cos(x) - 1)*sgn(sin(x))) - 2*log(-(cos(x) - 1)/
(cos(x) + 1))/(sqrt(a)*sgn(sin(x))) + (cos(x) - 1)/(sqrt(a)*(cos(x) + 1)*sgn(sin(x))))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^2\right )}^{3/2}} \,d x \]

[In]

int(1/(a*sin(x)^2)^(3/2),x)

[Out]

int(1/(a*sin(x)^2)^(3/2), x)